18.125 Homework 11
due Wed Apr 27 in class
1. (2 pts) Consider the following sequence of functions on R:
(
e−x , x ≥ 0;
f1 (x) =
0,
otherwise;
fn+1 = fn ∗ f1 ,
n = 1, 2, . . .
Derive the following formula for fn and conclude that fn+2 ∈ C n (R) for n = 0, 1, . . . :

−x n−1

e x
, x ≥ 0;
fn (x) = (n − 1)!

0,
otherwise.
(Remark: this is one example of how convolution tends to make functions more regular.)
2. (1 pt) Show that for f ∈ Lp (R), 1 ≤ p < ∞, the function
r Z
n
2
fn (x) =
f (y)e−n|x−y| dy
π R
converges to f in Lp (R) as n → ∞.
3. (2 pts) Let N ≥ 3. Fix R > 0, p ∈ [1, N2 ], and assume that f ∈ Lp (RN ) is equal to 0 outside the
ball B(0, R). Show that the integral
Z
f (y)
dy
(1)
u(x) =
|x
−
y|N −2
RN
converges absolutely for almost every x and for some constant C depending only on N, R, r,
Np
kukLr (B(0,R)) ≤ Ckf kLp (RN ) , 1 ≤ r <
.
N − 2p
If instead p >
N
2,
show that
sup |u| ≤ Ckf kLp (RN ) .
(Remark: the above bounds are a special case of Sobolev inequalities. The operator defined in (1)
is important since for f ∈ Cc∞ (RN ), u solves Poisson’s equation: ∆u = cN f for some constant cN .
1
For instance, when N = 3, 4π
u is the electric potential generated by the charge with density f .)
4. (2 pts) Fix α ∈ (1/2, 1) and consider the following function f on R:
(
|x|−α , 0 < |x| < 1;
f (x) =
0,
otherwise.
For which x does the following integral converge absolutely:
Z
f ∗ f (x) =
f (y)f (x − y) dy?
R
Show that f ∗ f ∈
Lp (R)
for all p ∈ [1,
1
2α−1 ).
5. (1 pt) Do Exercise 6.3.16.
6. (1 pt) Do Exercise 6.3.18 (i).
7. (1 pt) Do Exercise 6.3.19.
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